Radiation Transport Calculations by a Monte Carlo Method. Hideo Hirayama and Yoshihito Namito KEK, High Energy Accelerator Research Organization

Transcription

1 Radiation Transport Calculations by a Monte Carlo Method Hideo Hirayama and Yoshihito Namito KEK, High Energy Accelerator Research Organization

2 Monte Carlo Method A method used to solve a problem with random numbers is called a Monte Carlo Method. A name of Monte Carlo was introduced by J. von Neumann and S. M. Ulam. Random numbers are key tool for the Monte Carlo method

3 Random numbers Use a dice, a roulette etc. very slow. Use a table of random numbers. A table of random numbers has been well examined concerning its statistical characteristics. It is required to store a whole table in computer data storage. It is currently is not very fast to produce random numbers. Use physical random numbers like the decay of a radioisotope. It is not easy to digitalize, and has a weakness concerning stability and reproducibility.

4 Pseudo random Numbers Produce random numbers successively from a seed random number, R 0, using a recurrence formula in the form of R n+1 =f(r n ) Pseudo random number residuals by a divider, m. There are m different integers at most and, therefore, pseudo random numbers have a limited period. God pseudo random numbers Fast to create A long sequence Reproducibility Good statistical characteristics It is possible to create pseudo random number between 0 and 1 by dividing by m.

5 Another Type of Random Number Generator RANECU random number generator F. James, A review of pseudorandom number generators, Comp. Phys Comm 60 (1990) Periodicity Can be easily controlled by 2 seeds (iseed1 and iseed2) Use as the random number generator of egs5 Marasaglia-Zaman random number generator G. Masaglia and A. Zaman, A New Class of Random Number Generator, Annals of Applied Probability 1(1991) Long periodicity ~ Portable to all 32-bit machines

6 Pseudo Random Number Linear congruence methods proposed by D. H. Lehmer is most widely used. R n+1 =mod(ar n +b,m) a, b and m are positive integer and m is the length of the integer value allowed in the compiler. Name a b m RANDU SLAC RAN SLAC RAN

7 Production of pseudo random numbers using pocket calculator Produce 10 random numbers for R 0 =3, a=5, b=0 and m=16. Confirm that the same random sequence appears from some point. What is a sequence in this case? Check for a different R 0.

8 n R n R n *5 R n+1 =mod(r n *a,m) 0 R 0 =3 15 R 1 =15-INT(15/16)*16= mod(r n *a,m)= R n *a-int(rn*a/m) INT(15/16)=0

9 Calculation π using random numbers Select 2 random numbers from an arbitrary place in Table 2 (created using SLAC RAN6) in order Count the number of pair which satisfy R 2 2 = ξ + η 1.0 AREA= π /4 π= 4 x (probability within this area)

10 Discrete probability process If a probability variable (x) takes on discrete value (x i ) with probabilities (p i ) such that x=x i if Cumulative probability function (PDF) 1, ) ( 1 = = = n i x n p i F + = + = = < = ) ( ) ( i j j i i j j i p x F p x F η

11 Discrete type sampling(1) Example) Sample reaction when photoelectric is 30%, Compton scattering 50% and pair production 20% Photoelectric Compton scattering Pair creation

12 Discrete type sampling (2) Random number η Probability of Compton scattering Photoelectric Compton Scattering Pair creation Cumulative probability function (pdf)

13 Example determination of photon reaction The probability of the photoelectric effect, Compton scattering and pair creation at a photon interaction are P photo, P Compt and P pair, respectively. P photo +P Compt + P pair =1 η P photo, photoelectric P photo η < Pphoto + PCompt, Compton scattering P photo P η, pair creation + Compt

14 Sampling method (direct sampling) A probability distribution function (PDF:f(x)) is defined over the range [a,b]. b ( f ( x) dx = 1) Its cumulative probability function (CDF:F(x)) a a x F ( x) = f ( xi ) dx By its definition, we can map F(x) onto a range of random variable, η, where 0 η 1. Having mapped the random number onto F(x), we may invert the equation to give x = F 1 ( η) The way to determine x by solving the above equation is called a direct method. i

15 F(x) 1 η 0 a x b x

16 Example-determination of flight distance If the interaction probability of a particle per unit distance is Σ t, the probability that a first interaction occurs between l and l+dl is p( l) dl = e Σ l t Σ t dl η = P( l) = 1 l = Σ t l 0 p( l 1 ) dl = 1 e ln(1 η) = λ ln(1 η) l : flight distance λ: mean free path 1 η is equivalent to η 1 Σ l t l = λ ln( η)

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18 Red:in User Code Black:egs5 Infinite geometry Photon Initial Condition:Energy Position Direction E 0, X 0, Y 0,Z 0, U 0, V 0, W 0

19 Determine Interaction Point l=-ln(δ)/µ x=x 0 +u 0 l, y=y 0 +v 0 l. z=z 0 +w 0 l l Initial Condition:Energy Position Direction e 0, x 0, y 0, z 0, u 0, v 0, w 0

20 Determine Interaction Type Photoelectric:a, Compton scattering:b, Pair creation:c δ<=a/(a+b+c):photoelectric a/(a+b+c)< δ<=(a+c)/(a+b+c):compton scattering δ>(a+c)/(a+b+c):pair creation x=x 0 +u 0 l, y=y 0 +v 0 l. z=z 0 +w 0 l l Initial Condition:Energy Position Direction e 0, x 0, y 0, z 0, u 0, v 0, w 0

21 Electron Follows produced particle Determination of energy and direction of each particle Photon l x=x 0 +u 0 l, y=y 0 +v 0 l. z=z 0 +w 0 l Initial Condition:Energy Position Direction e 0, x 0, y 0, z 0, u 0, v 0, w 0

22 d>l:move to Interaction point d<=l:move a distance d Region boundary Same material:distance to Interaction point=l-d Different material:determine Interaction point Calculate a straight-line distance to the boundary (HOWFAR) l Interaction point: x=x 0 +u 0 l, y=y 0 +v 0 l. z=z 0 +w 0 l d Initial Condition:Energy Position Direction e 0, x 0, y 0, z 0, u 0, v 0, w 0

23 Region boundary Calculate a straight-line distance to the boundary (HOWFAR) l Store Information (AUSGAB) Particle moves:energy deposition Track length Boundary crossing Photoelectric:Energy deposition Below cut-off IDISC=1 IAUSFL(IARG)=1:optional d Initial Condition:Energy Position Direction e 0, x 0, y 0, z 0, u 0, v 0, w 0

24 AUSGAB Region boundary Store Information HOWFAR Calculate a straight-line distance to the boundary Set IDISCT=1 l d Main program Material assignment, Geometry Initial condition:energy Position Direction E 0, X 0, Y 0,Z 0, U 0, V 0, W 0 Analysis of results and output them

25 Determine Interaction Point l=-ln(δ)/σ It is difficult to treat an elastic scattering of electrons (or positrons) like photons due to its large number. l 5 x l 6 Charged particle lose a part of its l 4 x kinetic energy when it moves some x x l distance via ionization or excitation 3 l Energy deposition at each step Electron and positron True track length x Stopping power (de/dx) Initial Condition:Energy Position Direction e 0, x 0, y 0, z 0, u 0, v 0, w 0 l x x l 7 l 8 x Condensed History Technique Divide flight path to many small steps and apply Multiple Scattering Model to obtain true track length and change of direction and displacement due to many elastic scattering

26 Photon transport using a pocket calculator (Fig.1) Suppose that material A having thickness of 50 cm exist as shown Fig MeV photons enter this system from the left end, the mean free path is 20 cm, the ratio of the photoelectric effect and Compton scattering is 1:1 and, a scattered photon does not change its energy or direction. Exp.1 First random number λ=-20.0ln(0.2336)= (cm)<50.0(cm) Next random number is (<0.5). photoelectric effect Exp. 2 Next random number is λ=-20.0ln( )= (cm)<50.0(cm) Next random number is (>0.5). Compton scattering Next random number is λ=-20.0ln( )= (cm)< (cm)

27 Single layer No. d(cm) Random l(cm) d>l d l Random Photo. Compt. number number Exp * * Exp * * * * * * *

28 Fig. 1 Trajectories for a single layer 0cm 10cm 20cm 30cm 40cm 50cm Exp. 1 Exp. 2 C C P C

29 Photon transport using a pocket calculator (Fig.2) Suppose that material B (10cm) exits behind material A (40cm) as shown in Fig MeV photons enter this system from the left end, the mean free path and the ratio of the photoelectric effect and Compton scattering in medium A are same as the previous case, the mean free path of medium B is 3cm, ratio of the photoelectric effect and Compton scattering of medium B is 3:1 and, a scattered photon does not change its energy or direction. Exp.1 First random number λ=-20.0ln( )= (cm)<40.0(cm) Next random number is (>0.5). Compton scattering Next random number is λ=-20.0ln(0.2336)= (cm)> (cm) Move to boundary (40.0cm) Next random number is λ=-3.0ln( )= (cm)<10.0(cm)

30 Medium A No. d(cm) Random number l(cm) d>l d l Random number Photo. Compt Exp * * * Medium B d(cm) Random number l(cm) d>l d l Random number Photo. Compt * * * * * *

31 Fig. 2 Trajectories in double layers. 0cm 10cm 20cm 30cm 40cm 50cm Ex. 1 C C C P Medium A Medium B

32 Complex, but more realistic, example of photon transport Consider 10cm aluminum plate shown in Fig.3. Suppose that 0.5 MeV photon enters this system from left end. Photon is scattered with equal probability for each 45 at Compton scattering for all photon energies. The photon energy after scattering is calculated by E = E0 E (1 cosθ )

33 Complex, but more realistic, example of photon transport Suppose that the azimuthal angle after Compton scattering is 0º or 180º with an equal probability. 0º is 90º left from the particle direction and 180º is 90º right. Use the mean free path (mfp) and branching ratio for each photon energy in Fig. 4 and 5. Set the cutoff energy of photons to 0.05 MeV.

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